Name:_________________________________Date:________________________
1.
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A. |
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B. |
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C. |
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D. |
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2. The first five terms of a sequence are given below.
83 , 71 , 59 , 47 , 35 , ...
What
is the next term of the sequence?
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A. |
25 |
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B. |
23 |
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C. |
22 |
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D. |
24 |
3. Look at the pattern below.
50, 100, 200, 400, ...
Which
number sentence can be used to determine n, the
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A. |
n = 25 × 92 |
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B. |
n = (25 × 9) - 1 |
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C. |
n = 25 × 29 |
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D. |
n = 25 ÷ 9 |
4. The first three patterns of a sequence of dots is shown below.
If the pattern continues indefinitely, which of the following expressions can
be used to determine the number of dots in the nth term of
the sequence?
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A. |
n(n + 3) |
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B. |
n(n + 1) |
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C. |
3n |
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D. |
n2 + 3 |
5.
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A. |
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B. |
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C. |
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D. |
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6. The first four terms of a sequence are given below.
10, 8, 6, 4, ...
What
is the eighth term of the sequence?
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A. |
4 |
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B. |
-6 |
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C. |
-2 |
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D. |
-4 |
7. Which of the following patterns is determined from the expression below?
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A. |
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B. |
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C. |
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D. |
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8. The first three patterns of a sequence of dots are shown below.
If the pattern continues indefinitely, which of the following expressions can
be used to determine the number of dots in the nth term of
the sequence?
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A. |
3n |
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B. |
n(n + 1) |
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C. |
n(n + 2) |
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D. |
n2 + 2 |
9. What is the eleventh term of the pattern below?
23, 32, 47, 68, 95, ...
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A. |
433 |
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B. |
383 |
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C. |
343 |
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D. |
86 |
10.
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A. |
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B. |
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C. |
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D. |
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11. The first five terms of a sequence are given below.
20 , 27 , 34 , 41 , 48 , ...
Determine
which of the following formulas gives the nth term of this
sequence.
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A. |
27
- 7n |
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B. |
13
+ 7n |
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C. |
26
- 6n |
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D. |
14
+ 6n |
12. Which expression can be used to determine the nth term in the pattern below?
-5, -26, -61, -110, -173, ...
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A. |
7n2
- 2 |
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B. |
-7n
+ 2 |
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C. |
-5n
- 7 |
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D. |
-7n2
+ 2 |
13. The first five terms of a sequence are given below.
18 , 22 , 26 , 30 , 34 , ...
What
is the next term of the sequence?
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A. |
36 |
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B. |
37 |
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C. |
38 |
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D. |
39 |
14. What is the sixth term in the pattern below?
19, 10, 1, -8, ...
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A. |
-159 |
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B. |
-26 |
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C. |
-114 |
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D. |
-17 |
15. What is the eighth term of the pattern below?
16, 35, 54, 73, ...
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A. |
49 |
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B. |
155 |
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C. |
149 |
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D. |
1,213 |
16. All points from which of the following patterns would be
contained on the given graph?
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A. |
-1
, 1 , 3 , 5 , ... |
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B. |
-7
, -11 , -15 , -19 , ... |
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C. |
-5
, -7 , -9 , -11 , ... |
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D. |
-6
, -9 , -12 , -15 , ... |
17. The first three patterns of a sequence of dots are shown below.
If the pattern continues indefinitely, which of the following expressions can
be used to determine the number of dots in the nth term of
the sequence?
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A. |
n(n + 3) |
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B. |
n(n + 5) |
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C. |
3n |
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D. |
n2 + 3 |
18. The first five terms of a sequence are given below.
39 , 43 , 47 , 51 , 55 , ...
Determine
which of the following formulas gives the nth term of this
sequence.
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A. |
35
+ 4n |
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B. |
43
- 4n |
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C. |
36
+ 3n |
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D. |
42
- 3n |
19. Which expression represents the total volume of the pictures shown if each cube has a side length of e?
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A. |
c3 + e3 |
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B. |
c3 · e |
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C. |
c · e3 |
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D. |
c3 · e3 |
20. Which of the following patterns is represented by the graph below?
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A. |
n2 - 4n - 3 |
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B. |
n2 + 5n - 3 |
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C. |
2n2
+ n + 4 |
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D. |
n2 - 5n - 3 |
Answers
1. A
2. B
3. C
4. D
5. C
6. D
7. A
8. C
9. B
10. D
11. B
12. D
13. C
14. B
15. C
16. C
17. A
18. A
19. D
20. D
Explanations
1.
2. First, find the difference between terms.
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71 - 83 |
=
-12 |
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59 - 71 |
=
-12 |
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47 - 59 |
=
-12 |
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35 - 47 |
=
-12 |
Since the difference, -12, is constant, subtract 12 from the fifth term to find
the next term.
35 - 12 = 23
3. The sequence of numbers given starts with 50. Look for a common difference.
There is no common difference, so look for a common ratio.
There is a common ratio of 2, therefore this sequence is geometric.
Look at the sequence term by term and find a pattern in terms of number
position.
It appears that the jth term will
equal 25 × 2j.
Therefore, the 9th number in the pattern will be 25 × 29.
4. Look at the sequence term by term and find the pattern in terms of the number position.
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n = 1, |
#
of dots = |
4 |
=
12 + 3 or 1 + 3 |
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n = 2, |
#
of dots = |
7 |
=
22 + 3 or 4 + 3 |
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n = 3, |
#
of dots = |
12 |
=
32 + 3 or 9 + 3 |
From looking at the patterns listed above, the number of dots is given by n2
+ 3.
5.
6. First, find the difference between terms.
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8 - 10 |
=
-2 |
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6 - 8 |
=
-2 |
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4 - 6 |
=
-2 |
Since the difference, -2, is constant, subtract 2 from the fourth, fifth,
sixth, and seventh terms to find the eighth term.
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4 - 2 |
=
2 |
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2 - 2 |
=
0 |
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0 - 2 |
=
-2 |
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-2 - 2 |
=
-4 |
Therefore, the eighth term of the sequence is -4.
7. Substitute n = 1, 2, 3, 4, ... into the expression to find the terms of the pattern.
\%20&=&\%20\frac%7b15%7d%7b28%7d\\%20\vspace*%7b9%20\hspace*%7b-10%7d%7d\\\frac%7b1%7d%7b4%7d\%20+\%20\frac%7b2%7d%7b7%7d(2)\%20&=&\%20\frac%7b23%7d%7b28%7d\\%20\vspace*%7b9%20\hspace*%7b-10%7d%7d\\\frac%7b1%7d%7b4%7d\%20+\%20\frac%7b2%7d%7b7%7d()
Therefore, the pattern determined from the expression is
shown below.
8. Look at the sequence term by term and find a pattern in terms of the number position.
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n = 1, |
#
of dots = |
3 |
=
1 × (1 + 2) |
or
1 × 3 |
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n = 2, |
#
of dots = |
8 |
=
2 × (2 + 2) |
or
2 × 4 |
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n = 3, |
#
of dots = |
15 |
=
3 × (3 + 2) |
or
3 × 5 |
From looking at the patterns listed above, the number of dots is given by
Thus, the number of dots in the nth term of the sequence is
given by
9. The first and second differences of the sequence are shown below.
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23 |
32 |
47 |
68 |
95 |
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\ |
/ |
\ |
/ |
\ |
/ |
\ |
/ |
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9 |
15 |
21 |
27 |
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\ |
/ |
\ |
/ |
\ |
/ |
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6 |
6 |
6 |
Since the second difference is constant, the pattern is quadratic with the form
an2 + bn + c.
Set the second difference equal to 2a, and solve for a.
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2a |
= |
6 |
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a |
= |
3 |
Next, set the first term in the first difference equal to 3a + b,
and solve for b.
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3a + b |
= |
9 |
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3(3) + b |
= |
9 |
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9 + b |
= |
9 |
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b |
= |
0 |
Finally, set the first term in the pattern equal to a + b + c,
and solve for c.
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a + b + c |
= |
23 |
|
3 + 0 + c |
= |
23 |
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c |
= |
20 |
Thus, the sequence can be represented by 3n2 + 20.
Therefore, the eleventh term of the sequence is 3(11)2 + 20 = 383.
10.
11. A generic arithmetic sequence is of the following form,
a , a + d , a + 2d , a + 3d , ... , a + (n - 1)d , ...
where a is the first term, d
is the common difference, and a + (n - 1)d is the nth
term.
In this case, the first term is 20 and the common difference is 7.
Therefore, the nth term is as follows.
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nth term = |
a + (n - 1)d |
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= |
20
+ (n - 1)(7) |
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= |
20
+ (7n - 7) |
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= |
13
+ 7n |
12. The first and second differences of the sequence are shown below.
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-5 |
-26 |
-61 |
-110 |
-173 |
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/ |
\ |
/ |
\ |
/ |
\ |
/ |
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-21 |
-35 |
-49 |
-63 |
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\ |
/ |
\ |
/ |
\ |
/ |
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-14 |
-14 |
-14 |
Since the second difference is constant, the pattern is quadratic with the form
an2 + bn + c.
Set the second difference equal to 2a, and solve for a.
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2a |
= |
-14 |
|
a |
= |
-7 |
Next, set the first term in the first difference equal to 3a + b,
and solve for b.
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3a + b |
= |
-21 |
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3(-7) + b |
= |
-21 |
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-21 + b |
= |
-21 |
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b |
= |
0 |
Finally, set the first term in the pattern equal to a + b + c,
and solve for c.
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a + b + c |
= |
-5 |
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-7 + 0 + c |
= |
-5 |
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c |
= |
2 |
Therefore, the sequence can be represented by -7n2 + 2.
13. First, find the difference between terms.
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22 - 18 |
=
4 |
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26 - 22 |
=
4 |
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30 - 26 |
=
4 |
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34 - 30 |
=
4 |
Since the difference, 4, is constant, add 4 to the fifth term to find the next
term.
34 + 4 = 38
14. First, find the difference between consecutive terms.
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10 - 19 |
= |
-9 |
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1 - 10 |
= |
-9 |
|
-8 - 1 |
= |
-9 |
The difference between consecutive terms is -9.
Subtract 9 from -8 to find the fifth term, and then subtract 9 from the fifth
term to find the sixth term.
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-8 - 9 |
= |
-17 |
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-17 - 9 |
= |
-26 |
15. First, find an expression that represents
the nth term of the pattern.
The pattern starts with 16 and increases by 19 each time. Look at the sequence
term by term to see if there is a recognizable pattern in terms of the position
of the number in the sequence.
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1st
term: |
16 |
= |
19
· 1 - 3 |
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2nd
term: |
35 |
= |
19
· 2 - 3 |
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3rd
term: |
54 |
= |
19
· 3 - 3 |
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4th
term: |
73 |
= |
19
· 4 - 3 |
Therefore, the expression 19n - 3 represents the nth
term in the pattern.
Substitute n = 8 into the expression and evaluate to find the eighth
term of the pattern.
19(8) - 3 = 149
16. The common difference in a linear pattern is equivalent to the slope of
the correlating line.
In the graph above, the slope is -2.
Therefore, the common difference of the correlating pattern is also -2.
The only answer choice with a common difference of -2 is
-5 , -7 , -9 , -11 , ....
17. Look at the sequence term by term and find a pattern in terms of number position.
|
n = 1, |
#
of dots = |
4 |
=
1 × (1 + 3) |
or
1 × 4 |
|
n = 2, |
#
of dots = |
10 |
=
2 × (2 + 3) |
or
2 × 5 |
|
n = 3, |
#
of dots = |
18 |
=
3 × (3 + 3) |
or
3 × 6 |
From looking at the patterns listed above, the number of dots is given by
Thus, the number of dots in the nth term of the sequence is
given by
18. A generic arithmetic sequence is of the following form,
a , a + d , a + 2d , a + 3d , ... , a + (n - 1)d , ...
where a is the first term, d
is the common difference, and a + (n - 1)d is the nth
term.
In this case, the first term is 39 and the common difference is 4.
Therefore, the nth term is as follows.
|
nth term = |
a + (n - 1)d |
|
= |
39
+ (n - 1)(4) |
|
= |
39
+ (4n - 4) |
|
= |
35
+ 4n |
19. The volume for each solid can be determined by multiplying the number of
cubes by the volume of each individual cube. Since the cubes each have a side
length of e, the volume of each cube is e3.
Looking at the pattern, the number of cubes in each picture is always equal to c3.
Therefore, the total volume is given by c3 · e3.
20. Substitute values of n into each
pattern to find which gives the same coordinates as those on the graph.
The pattern n2 - 5n - 3 matches the graph as shown
below.
|
n |
n2 - 5n - 3 |
(n,
an) |
|
0 |
(0)2
- 5(0) - 3 = -3 |
(0,-3) |
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1 |
(1)2
- 5(1) - 3 = -7 |
(1,-7) |
|
2 |
(2)2
- 5(2) - 3 = -9 |
(2,-9) |
|
3 |
(3)2
- 5(3) - 3 = -9 |
(3,-9) |
|
4 |
(4)2
- 5(4) - 3 = -7 |
(4,-7) |
|
5 |
(5)2
- 5(5) - 3 = -7 |
(5,-3) |
|
6 |
(6)2
- 5(6) - 3 = -7 |
(6,
3) |
Therefore, the graph represents the pattern n2 - 5n
- 3 because the pattern gives the same coordinates as those on the graph.